Unconditional error analysis of the linearized transformed L1 virtual element method for nonlinear coupled time-fractional Schrödinger equations

Abstract

This paper constructs a linearized transformed $L1$ virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical $s$-fractional differential system derived from a smoothing transformation of variables $t=s^{1/\alpha }$, $0<\alpha <1$. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in $L^2$-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.